Unlike the familiar
L
p
{L^p}
spaces, neither the homogeneous Besov spaces nor the
H
p
{H^p}
spaces,
0
>
p
>
1
0\, > \,p\, > \,\,1
, are closed under multiplication by the functions
x
→
e
i
⟨
x
,
h
⟩
x\, \to \,{e^{i\left \langle {x,h} \right \rangle }}
. We determine the maximal subspace of these spaces which are closed under multiplication by these functions, which are the characters of
R
n
{R^n}
.